\subsection{Tony's comment}
I should look into at either end, how much is the ratio of close/open and then establish the relation between \cite{JacksonNarrow} and Tony's. 

However, it is still not clear to me how to really map broad ($\Delta\mu\Delta{}B\gg\frac{k_F}{a_{bg}}$)/narrow resonance into the many-body eq.(\ref{eq:renomal1}) in a calculable sense.  